I am creating a logo in Photoshop and I am going to use it for the blog site I am trying to develop. I plan to make the website responsive.
I think I should use .png, but I don't know if that is the best option. What file type should I use and why?
Yes, PNG is just fine.
The vector part is not necesary when you deliver the website, but when you create it (1).
If the logo is not very complex, you can use Inkscape to draw it again, and you could use the native SVG.
But if your logo has effects like shadows, it is ok on PNG.
(1) Normally you need to create a logo on vectors, becouse it is suposed that you need flexibility. You need diferent sizes, diferent color aplications, you probably will need to send it to make Tshirts or a mug. You normally do not create your logo on Photoshop.
You can add effects later on photoshop, shadows, sparkles and happy minions if you need, but the base logo it is recomended that is on vectors.
Logos should generally be designed and executed in Illustrator, so they can be vector images.
You can create a vector in Photoshop, but you should not create a raster logo, with pixels. The reason is that when a raster logo is scaled up larger, it gets blurry, pixelated and looks bad.
Photoshop is clumsy with vectors, while Illustrator is good at them. However, if you only have Photoshop, you can make it work. But create a vector logo - not a raster one.